Let $\varphi:(A,\mathfrak{m})\to(B,\mathfrak{n})$ be an integral extension of regular local rings of dimension $d$ (of course, $\varphi$ is a local homomorphism). Furthermore, assume that $A$ contains the field $\mathbb{C}$ of complex numbers. If $\mathfrak{m}=(x_1,\ldots,x_d)$ is a regular system of parameters of $A$, is it possible to choose a regular system of parameters $\mathfrak{n}=(y_1,\ldots,y_d)$ such that there exist integers $n_i$ with $\varphi(x_i)=y_i^{n_i}$?